Twitter is a platform that I have not got into, but clearly it is very popular and powerful.PostGhost is an archive of public tweets deleted by politicians, celebrities, and other public figures. On July 6, 2016, PostGhost.com received an email from Twitter, which says in part that "postghost.com displays deleted Tweets and is currently violating their Developer Agreement and Policy".

In the days leading up to and during the UK’s EU referendum, many major celebrities used Twitter to broadcast their political agenda to their hundreds of thousands or millions of followers (Such as Johnny Robinson, singer with 151K followers, J.K. Rowling, author with 7.6M followers, and Lindsay Lohan, actress with 9.3M followers).

These three tweets, alternately urging people to #leave or #remain, were sent out to a collective 17 million followers – a reach exceeding practically every politician on Twitter. Many followers received them on their phones instantly. All three tweets were then deleted within minutes. For non-followers and people who don’t use Twitter, it’s as if the tweets never existed, and no record of them exists aside from PostGhost. In a referendum decided by just over 1 million votes, the ability to reach millions of followers instantly and leave no trace is a massive and growing power, and one that is currently completely unchecked and undocumented.

I think they make good points, and on the face of it I think somebody probably should be doing it, but I am not clear why they necessarily need Twitter's approval to do it, although clearly they do to do it the way they have been.So the question is: if they find another way to archive information posted publicly, is there a legal way to make them stop?

It would appear to be a copyright infringement. You do own a tweet. When you publish it, you publish it under twitter terms and conditions. I have not read them, but commonly they will at the very least take for themselves a licence to reproduce a tweet. There will be additional complex licences and restrictions.The fact that a tweet is reproduced a hundred thousand times does not make it public domain. A person is free to do whatever acts are compatible with his ownership of the copyright, including, by default, the right to withdraw a tweet. If it did nothing to support its terms, it would risk undermining its own business model, possibly also leaving itself open to legal action from its twitterers.

If indeed the action is taken within Twitter's terms, I cannot see that the site has a complaint valid in law - though of course this would be US law, not British.